Symmetric and alternating-squares of linear representation: Difference between revisions

Latest revision as of 10:49, 18 November 2023

This article gives a basic definition in the following area: linear representation theory
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Definition

Let $(ρ, V)$ be a linear representation of a group $G$ . Then we can define the linear representations $(S^{2} ρ, S^{2} V)$ and $(Λ^{2} ρ, Λ^{2} V)$ , the symmetric and alternating-squares of $(ρ, V)$ respectively, by restricting the representation $(ρ \otimes ρ, V \otimes V)$ of $G$ to the eigenspaces corresponding to the symmetric and alternating-squares of a vector space respectively, that is,

$S^{2} ρ (g) (v \otimes w) = ρ (g) (v) \otimes ρ (g) (w)$ for $v \otimes w \in S^{2} V$ , $Λ^{2} ρ (g) (v \otimes w) = ρ (g) (v) \otimes ρ (g) (w)$ for $v \otimes w \in Λ^{2} V$ .

Facts

Even if $ρ$ is irreducible, $S^{2} ρ$ , $Λ^{2} ρ$ need not be irreducible. See the example given in this article.

Characters of the symmetric and alternating-squares

For a representation $ρ$ , write $χ_{ρ}$ for its character.

Then

$χ_{S^{2} ρ} = \frac{1}{2} (χ_{ρ} (g)^{2} + χ_{ρ} (g^{2}))$ , and

$χ_{Λ^{2} ρ} = \frac{1}{2} (χ_{ρ} (g)^{2} - χ_{ρ} (g^{2}))$ .

Example

Dihedral group of order 8

Further information: linear representation theory of dihedral group:D8

Consider the dihedral group:D8 $⟨ x, a | a^{4} = x^{2} = e, x a x^{- 1} = a^{- 1} ⟩$ which has a two-dimensional faithful irreducible representation given by

$ρ (a) = (\begin{matrix} 0 & - 1 \\ 1 & 0 \end{matrix}), ρ (x) = (\begin{matrix} 1 & 0 \\ 0 & - 1 \end{matrix})$ .

(This specifies the whole representation as representations are group homomorphisms, which are specified by their action on generators of a group.)

Calculating $S^{2} ρ$ gives a non-trivial three-dimensional representation of the group:

$S^{2} ρ (a) = (\begin{matrix} 0 & 0 & 1 \\ 0 & - 1 & 0 \\ 1 & 0 & 0 \end{matrix}), S^{2} ρ (x) = (\begin{matrix} 1 & 0 & 0 \\ 0 & - 1 & 0 \\ 0 & 0 & 1 \end{matrix})$ ,

which is reducible.

Calculating $Λ^{2} ρ$ gives a non-trivial one-dimensional representation of the group:

$Λ^{2} ρ (a) = (\begin{matrix} 1 \end{matrix}), Λ^{2} ρ (x) = (\begin{matrix} - 1 \end{matrix})$ .

Applications

Character tables

When calculating the character table of a group, if some irreducible representations are unknown, we may find a (not necessarily irreducible) representation by taking the symmetric or alternating squares of found representations. Even if this is not irreducible, it can be decomposed into a direct sum of irreducible subrepresentations, some of which may be new, allowing for more rows in the character table to be filled.

An example of when this occurs is with the symmetric group:S4. A "natural/human" way to start is by writing down the trivial, sign and standard representations. The rest can be derived by tensor products and calculating alternating-squares. See linear representation theory of symmetric group:S4, Determination of character table of symmetric group:S4

@@ Line 3: / Line 3: @@
 ==Definition==
-Let <math>(\rho, V)</math> be a [[linear representation]] of a [[group]] <math>G</math>. Then we can define the linear representations <math>(S^2 \rho, S^2 V)</math> and <math>(\Lambda^2 \rho, \Lambda^2 V)</math>, the '''symmetric and alternating-squares''' of <math>(\rho, V)</math> respectively, by restricting the representation <math>(\rho \otimes \rho, V \otimes V)</math> of <math>G</math> to the eigenspaces corresponding to the symmetric and alternating-squares respectively, that is,
+Let <math>(\rho, V)</math> be a [[linear representation]] of a [[group]] <math>G</math>. Then we can define the linear representations <math>(S^2 \rho, S^2 V)</math> and <math>(\Lambda^2 \rho, \Lambda^2 V)</math>, the '''symmetric and alternating-squares''' of <math>(\rho, V)</math> respectively, by restricting the representation <math>(\rho \otimes \rho, V \otimes V)</math> of <math>G</math> to the eigenspaces corresponding to the [[symmetric and alternating-squares of a vector space]] respectively, that is,
-<math>S^2 \rho(g) (v \otimes w) = \rho(g)(v) \otimes \rho(g) w</math> for <math>v \otimes w \in S^2 V</math>, <math>\Lambda^2 \rho(g) (v \otimes w) = \rho(g)(v) \otimes \rho(g) w</math> for <math>v \otimes w \in \Lambda^2 V</math>.
+<math>S^2 \rho(g) (v \otimes w) = \rho(g)(v) \otimes \rho(g)(w)</math> for <math>v \otimes w \in S^2 V</math>, <math>\Lambda^2 \rho(g) (v \otimes w) = \rho(g)(v) \otimes \rho(g)(w)</math> for <math>v \otimes w \in \Lambda^2 V</math>.
+==Facts==
+* Even if <math>\rho</math> is [[irreducible representation|irreducible]], <math>S^2 \rho</math>, <math>\Lambda^2 \rho</math> need not be irreducible. See the example given in this article.
 ==Characters of the symmetric and alternating-squares==
@@ Line 16: / Line 20: @@
 <math>\chi_{\Lambda^2 \rho} = \frac{1}{2} (\chi_{\rho}(g)^2 -\chi_{\rho}(g^2))</math>.
+==Example==
+===Dihedral group of order 8===
+{{further|[[linear representation theory of dihedral group:D8]]}}
+Consider the [[dihedral group:D8]] <math>\langle x,a| a^4 = x^2 = e, xax^{-1} = a^{-1}\rangle</math> which has a two-dimensional faithful [[irreducible representation|irreducible]] representation given by
+<math>\rho(a) = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}, \rho(x) = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}</math>.
+(This specifies the whole representation as representations are [[group homomorphism|group homomorphisms]], which are specified by their action on generators of a group.)
+Calculating <math>S^2 \rho</math> gives a non-trivial three-dimensional representation of the group:
+<math>S^2 \rho(a) =  \begin{pmatrix} 0 & 0 & 1 \\
+& -1 & 0 \\ 1 & 0 & 0 \end{pmatrix}, S^2 \rho(x) = \begin{pmatrix} 1 & 0 & 0 \\
+& -1 & 0 \\ 0 & 0 & 1\end{pmatrix}</math>,
+which is reducible.
+Calculating <math>\Lambda^2 \rho</math> gives a non-trivial one-dimensional representation of the group:
+<math>\Lambda^2 \rho(a) = \begin{pmatrix} 1 \end{pmatrix}, \Lambda^2 \rho(x) = \begin{pmatrix} -1 \end{pmatrix}</math>.
+==Applications==
+===Character tables===
+When calculating the [[character table]] of a group, if some irreducible representations are unknown, we may find a (not necessarily irreducible) representation by taking the symmetric or alternating squares of found representations. Even if this is not irreducible, it can be decomposed into a direct sum of irreducible subrepresentations, some of which may be new, allowing for more rows in the character table to be filled.
+An example of when this occurs is with the [[symmetric group:S4]]. A "natural/human" way to start is by writing down the trivial, sign and standard representations. The rest can be derived by tensor products and calculating alternating-squares. See [[linear representation theory of symmetric group:S4]], [[Determination of character table of symmetric group:S4]]